Structures of SU-rank omega with a dense independentsubset of generics

Extending the work done in \cite{BV-Tind,DMS} in the o-minimal and geometric settings, we study expansions of models of a supersimple theory of SU-rank $\omega$ with a "dense codense" independent collection$H$ of element of rank $\omega$, where density of $H$ means it intersectsany definable set of $SU$-rank omega. We show that under some technical conditions, the class of such structures is first order.We prove that the expansion is supersimple and characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one-basedness and CM-triviality. In the one-based case, we describe a natural "geometry of generics modulo $H$" associated with such expansions and show it is modular